This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A325691 #33 May 19 2023 11:02:37
%S A325691 0,0,0,1,1,1,2,2,3,3,4,4,6,5,7,7,9,8,11,10,13,12,15,14,18,16,20,19,23,
%T A325691 21,26,24,29,27,32,30,36,33,39,37,43,40,47,44,51,48,55,52,60,56,64,61,
%U A325691 69,65,74,70,79,75,84,80,90,85,95,91,101,96,107,102,113
%N A325691 Number of length-3 integer partitions of n whose largest part is not greater than the sum of the other two.
%C A325691 Also the number of possible triples of edge-lengths of a triangle with perimeter n, where degenerate (self-intersecting) triangles are allowed.
%C A325691 The number of triples (a,b,c) for 1 <= a <= b <= c <= a+b and a+b+c = n. - _Yuchun Ji_, Oct 15 2020
%H A325691 Stefano Spezia, <a href="/A325691/b325691.txt">Table of n, a(n) for n = 0..10000</a>
%H A325691 Colin Defant, Michael Joseph, Matthew Macauley, and Alex McDonough, <a href="https://arxiv.org/abs/2305.07627">Torsors and tilings from toric toggling</a>, arXiv:2305.07627 [math.CO], 2023. See g.f. at p. 20.
%H A325691 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,-1,-1,-1,0,1).
%F A325691 Conjectures from _Colin Barker_, May 16 2019: (Start)
%F A325691 G.f.: x^3*(1 + x - x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
%F A325691 a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8. (End)
%F A325691 a(n) = A005044(n+3) - A000035(n+3). i.e., remove the only one triple (a=0,b,b) if n is even from the A005044 which is the number of triples (a,b,c) for 0 <= a <= b <= c <= a+b and a+b+c = n. - _Yuchun Ji_, Oct 15 2020
%F A325691 The above conjectured formulas are true. - _Stefano Spezia_, May 19 2023
%e A325691 The a(3) = 1 through a(12) = 6 partitions:
%e A325691 (111) (211) (221) (222) (322) (332) (333) (433) (443) (444)
%e A325691 (321) (331) (422) (432) (442) (533) (543)
%e A325691 (431) (441) (532) (542) (552)
%e A325691 (541) (551) (633)
%e A325691 (642)
%e A325691 (651)
%t A325691 Table[Length[Select[IntegerPartitions[n,{3}],#[[1]]<=#[[2]]+#[[3]]&]],{n,0,30}]
%Y A325691 Cf. A001399, A005044 (nondegenerate triangles), A008642, A069905, A124278.
%Y A325691 Cf. A325688, A325690, A325694.
%K A325691 nonn,easy
%O A325691 0,7
%A A325691 _Gus Wiseman_, May 15 2019